Asymptotic behavior of a hydrodynamic system in the nematic liquid crystal flows

In this paper we study the long time behavior of the classical solutions to a hydrodynamical system modeling the flow of nematic liquid crystals. This system consists of a coupled system of Navier--Stokes equations and kinematic transport equations for the molecular orientations. By using a suitable Lojasiewicz--Simon type inequality, we prove the convergence of global solutions to single steady states as time tends to infinity. Moreover, we provide estimates for the convergence rate